题目：Coadjoint orbits of Sternberg type and their geometric quantization

报告人：孟国武，香港科技大学

时间：1月8日（周三）15:00-16:00

地点：东区第五教学楼5206教室

摘要：Let $k\ge 1$ be an integer and $\mu$ be the half of a {\it nonzero} integer. The following statements hold for the elliptic co-adjoint orbit of the real Lie algebra $\mathfrak{so}(2, 2k+2)$ that corresponds to the dominant weight $(\underbrace{|\mu|, \ldots, |\mu|}_{k+1}, \mu)$.

1. This orbit is diffeomorphic to $\mathrm{SO}_0(2, 2k+2)/\mathrm{U}(1, k+1)$. As a result, it is pre-quantizable.

2. This orbit is the total space of a fiber bundle with base space being the total cotangent space of the punctured euclidean space of dimension $2k+1$ and the fiber being diffeomorphic to $\mathrm{SO}(2n)/\mathrm{U}(n)$. As a result, it admits a canonical polarization.

3. The geometric quantization of this orbit with its canonical polarization yields the Hilbert of square integrable sections of a Hermitian vector bundle over the punctured Euclidean space in dimension $2k+1$; moreover, this Hilbert space provides a geometric realization for the unitary highest weight $\frak{so}(2, 2k+2)$-module with highest weight \[(-k-|\mu|, \underbrace{ |\mu|, \ldots, |\mu|}_k, \mu).\]

The above results in Lie theory is obtained from the study of magnetized Kepler models in dimension $2k+1$. 

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